Method for designing an index profile suitable for encoding into a phase mask for manufacturing a complex optical grating

Abstract

A method for designing an index profile suitable for encoding into a phase mask for manufacturing a complex optical grating is provided. The optical grating corresponds to a target index profile defining a target spectral response. A modified index profile is set equal to the target index profile and expressed as a function of an apodization and phase profiles. The modified index profile is iteratively further modified in order to provide the index profile suitable for encoding in the phase mask. This process creates side bands outside of a spectral region of interest, while maintaining the target spectral response within the spectral region of interest.

1 . A method for designing an index profile suitable for encoding into a phase mask for manufacturing a complex optical grating, said optical grating corresponding to a target index profile having a Fourier spectrum and defining a target optical spectral response inside a spectral region of interest, the method comprising the steps of: 
 (a) setting a modified index profile equal to said target index profile;    (b) expressing said modified index profile as a function of a modified apodization profile and a modified phase profile;    (c) filtering the modified phase profile in order to minimize phase periodicities therein, thereby obtaining a filtered phase profile;    (d) replacing the modified apodization profile by a smooth apodization profile, and replacing the modified phase profile by the filtered phase profile;    (e) calculating a Fourier spectrum of the modified index profile obtained through the replacing of step (d);    (f) calculating a hybrid index profile having a Fourier spectrum corresponding to a sum of said Fourier spectrum of the modified index profile outside said spectral region of interest with said Fourier spectrum of said target index profile within said spectral region of interest;    (g) setting the modified index profile equal to the hybrid index profile; and    (h) repeating steps (b) through (g) using the modified index profile obtained at step (g) as a starting point until said index profile suitable for encoding into a phase mask is obtained.    
     
     
         2 . The method according to  claim 1 , wherein step (b) comprises expressing said modified index profile as:  
         Δ n   struct,mod ( z )=Δ n   offset   +Δn   m ( z )·exp( iθ   m ( z ))  
       wherein Δn struct,mod (z) is the modified index profile, Δn offset  is an index offset making the modified index profile strictly positive, Δn m (z) is the modified apodization profile, and θ m (z) is the modified phase profile.  
     
     
         3 . The method according to  claim 1 , wherein the filtering of step (c) comprises the substeps of: 
 i. calculating a phase shift profile associated with the phase profile;    ii. calculating a Fourier spectrum of said phase shift profile; and    iii. filtering said Fourier spectrum of the phase shift profile through a filter function.    
     
     
         4 . The method according to  claim 3 , wherein said filter function is set to 0 for spatial frequencies where said phase periodicities are present, and set to 1 otherwise.  
     
     
         5 . The method according to  claim 1 , wherein the smooth apodization profile of step (d) is set to a predetermined profile or to a spatially smoothed version of the modified apodization profile.  
     
     
         6 . The method according to  claim 1 , wherein the repeating of step (h) is carried out until the modified apodization profile is substantially free of fine structures and of phase periodicities.  
     
     
         7 . A method for manufacturing a phase mask for photoinducing a complex optical grating in a photosensitive medium, said complex optical grating corresponding to a target index profile having a Fourier spectrum and defining a target optical spectral response inside a spectral region of interest, the method comprising the steps of: 
 A) designing an index profile suitable for encoding into a phase mask, said designing comprising the sub-steps of: 
 (a) setting a modified index profile equal to said target index profile;  
 (b) expressing said modified index profile as a function of a modified apodization profile and a modified phase profile;  
 (c) filtering the modified phase profile in order to minimize phase periodicities therein, thereby obtaining a filtered phase profile;  
 (d) replacing the modified apodization profile by a smooth apodization profile, and replacing the modified phase profile by the filtered phase profile;  
 (e) calculating a Fourier spectrum of the modified index profile obtained through the replacing of sub-step (d);  
 (f) calculating a hybrid index profile having a Fourier spectrum corresponding to a sum of said Fourier spectrum of the modified index profile outside said spectral region of interest with said Fourier spectrum of said target index profile within said spectral region of interest;  
 (g) setting the modified index profile equal to the hybrid index profile; and  
 (h) repeating sub-steps (b) through (g) using the modified index profile obtained at sub-step (g) as a starting point until said index profile suitable for encoding into a phase mask is obtained; and  
   B) encoding a phase mask according to said index profile suitable for encoding.    
     
     
         8 . The method according to  claim 7 , wherein sub-step A)(b) comprises expressing said modified index profile as:  
         Δ n   struct,mod ( z )=Δ n   offset   +Δn   m ( z )·exp( iθ   m ( z ))  
       wherein Δn struct,mod (z) is the modified index profile, Δn offset  is an index offset making the modified index profile strictly positive, Δn m (z) is the modified apodization profile, and θ m (z) is the modified phase profile.  
     
     
         9 . The method according to  claim 7 , wherein the filtering of sub-step A)(c) comprises the substeps of: 
 iv. calculating a phase shift profile associated with the phase profile;    v. calculating a Fourier spectrum of said phase shift profile; and    vi. filtering said Fourier spectrum of the phase shift profile through a filter function.    
     
     
         10 . The method according to  claim 9 , wherein said filter function is set to 0 for spatial frequencies where said phase periodicities are present, and set to 1 otherwise.  
     
     
         11 . The method according to  claim 7 , wherein the smooth apodization profile of sub-step A)(d) is set to a predetermined profile or to a spatially smoothed version of the modified apodization profile.  
     
     
         12 . The method according to  claim 7 , wherein the repeating of sub-step A)(h) is carried out until the modified apodization profile is substantially free of fine structures and of phase periodicities.  
     
     
         13 . The method according to  claim 7 , wherein the encoding of step B) comprises manufacturing said phase mask with corrugations having a phase profile related to the index profile suitable for encoding through a transfer function.  
     
     
         14 . The method according to  claim 13 , wherein said transfer function is given by:  
           T= 2 cos(π dz f )  
       where f is a spatial frequency and dz is a distance between two points along the phase mask from which pass two beams interfering in the photosensitive medium at a position z.  
     
     
         15 . A method for manufacturing a complex optical grating in a photosensitive medium, said complex optical grating corresponding to a target index profile having a Fourier spectrum and defining a target optical spectral response inside a spectral region of interest, the method comprising the steps of: 
 A) designing an index profile suitable for encoding into a phase mask, said designing comprising the sub-steps of: 
 (a) setting a modified index profile equal to said target index profile;  
 (b) expressing said modified index profile as a function of a modified apodization profile and a modified phase profile;  
 (c) filtering the modified phase profile in order to minimize phase periodicities therein, thereby obtaining a filtered phase profile;  
 (d) replacing the modified apodization profile by a smooth apodization profile, and replacing the modified phase profile by the filtered phase profile;  
 (e) calculating a Fourier spectrum of the modified index profile obtained through the replacing of sub-step (d);  
 (f) calculating a hybrid index profile having a Fourier spectrum corresponding to a sum of the Fourier spectrum of the modified index profile outside said spectral region of interest with said Fourier spectrum of said target index profile within said spectral region of interest;  
 (g) setting the modified index profile equal to the hybrid index profile; and  
 (h) repeating sub-steps (b) through (g) using the modified index profile obtained at sub-step (g) as a starting point until said index profile suitable for encoding into a phase mask is obtained;  
   B) encoding a phase mask according to said index profile suitable for encoding; and    C) photoinducing said complex grating in the photosensitive medium using said phase mask.    
     
     
         16 . The method according to  claim 15 , wherein sub-step A)(b) comprises expressing said modified index profile as:  
         Δ n   struct,mod ( z )=Δ n   offset   +Δn   m ( z )·exp( iθ   m ( z ))  
       wherein Δn struct,mod (z) is the modified index profile, Δn offset  is an index offset making the modified index profile strictly positive, Δn m (z) is the modified apodization profile, and θ m (z) is the modified phase profile.  
     
     
         17 . The method according to  claim 15 , wherein the filtering of sub-step A)(c) comprises the substeps of: 
 vii. calculating a phase shift profile associated with the phase profile;    viii. calculating a Fourier spectrum of said phase shift profile; and    ix. filtering said Fourier spectrum of the phase shift profile through a filter function.    
     
     
         18 . The method according to  claim 17 , wherein said filter function is set to 0 for spatial frequencies where said phase periodicities are present, and set to 1 otherwise.  
     
     
         19 . The method according to  claim 15 , wherein the smooth apodization profile of sub-step A)(d) is set to a predetermined profile or to a spatially smoothed version of the modified apodization profile.  
     
     
         20 . The method according to  claim 15 , wherein the repeating of sub-step A)(h) is carried out until the modified apodization profile is substantially free of fine structures and of phase periodicities.  
     
     
         21 . The method according to  claim 15 , wherein the encoding of step B) comprises manufacturing said phase mask with corrugations having a phase profile related to the index profile suitable for encoding through a transfer function.  
     
     
         22 . The method according to  claim 21 , wherein said transfer function is given by:  
           T= 2 cos(π dz f )  
       where f is a spatial frequency and dz is a distance between two points along the phase mask from which pass two beams interfering in the photosensitive medium at a position z.  
     
     
         23 . The method according to  claim 15 , wherein the photoinducing of step C) comprises the substeps of: 
 i. disposing the photosensitive medium along the phase mask in close proximity thereto; and    ii. projecting actinic radiation through said phase mask, said actinic radiation being diffracted by said phase mask to generate the complex optical grating, said complex optical grating being photoinduced into the photosensitive medium. A method for manufacturing a phase mask using such a design and a method for manufacturing a complex optical grating using such a phase mask are also provided.